Theorem dcomex 9550

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 7808	. . . . . . 7 ⊢ 1𝑜 ≠ ∅
2		df-br 4845	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩})
3		elsni 4387	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩)
4		fvex 6417	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6417	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5133	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1𝑜, 1𝑜⟩ → (𝑓‘𝑛) = 1𝑜)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1𝑜, 1𝑜⟩} → (𝑓‘𝑛) = 1𝑜)
8	2, 7	sylbi 208	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1𝑜)
9		tz6.12i 6430	. . . . . . 7 ⊢ (1𝑜 ≠ ∅ → ((𝑓‘𝑛) = 1𝑜 → 𝑛𝑓1𝑜))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1𝑜)
11		vex 3394	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 7800	. . . . . . 7 ⊢ 1𝑜 ∈ V
13	11, 12	breldm 5530	. . . . . 6 ⊢ (𝑛𝑓1𝑜 → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3140	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3787	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 225	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3394	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7325	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 4997	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5098	. . 3 ⊢ {⟨1𝑜, 1𝑜⟩} ∈ V
23	12, 12	fvsn 6667	. . . . . . . 8 ⊢ ({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜
24	12, 12	funsn 6149	. . . . . . . . 9 ⊢ Fun {⟨1𝑜, 1𝑜⟩}
25	12	snid 4402	. . . . . . . . . 10 ⊢ 1𝑜 ∈ {1𝑜}
26	12	dmsnop 5821	. . . . . . . . . 10 ⊢ dom {⟨1𝑜, 1𝑜⟩} = {1𝑜}
27	25, 26	eleqtrri 2884	. . . . . . . . 9 ⊢ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}
28		funbrfvb 6454	. . . . . . . . 9 ⊢ ((Fun {⟨1𝑜, 1𝑜⟩} ∧ 1𝑜 ∈ dom {⟨1𝑜, 1𝑜⟩}) → (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
29	24, 27, 28	mp2an 675	. . . . . . . 8 ⊢ (({⟨1𝑜, 1𝑜⟩}‘1𝑜) = 1𝑜 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜)
30	23, 29	mpbi 221	. . . . . . 7 ⊢ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜
31		breq12 4849	. . . . . . . 8 ⊢ ((𝑠 = 1𝑜 ∧ 𝑡 = 1𝑜) → (𝑠{⟨1𝑜, 1𝑜⟩}𝑡 ↔ 1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜))
32	12, 12, 31	spc2ev 3494	. . . . . . 7 ⊢ (1𝑜{⟨1𝑜, 1𝑜⟩}1𝑜 → ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡
34		breq 4846	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
35	34	2exbidv 2015	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1𝑜, 1𝑜⟩}𝑡))
36	33, 35	mpbiri 249	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 3820	. . . . . . 7 ⊢ {1𝑜} ⊆ {1𝑜}
38	12	rnsnop 5829	. . . . . . 7 ⊢ ran {⟨1𝑜, 1𝑜⟩} = {1𝑜}
39	37, 38, 26	3sstr4i 3841	. . . . . 6 ⊢ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}
40		rneq 5552	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 = ran {⟨1𝑜, 1𝑜⟩})
41		dmeq 5525	. . . . . . 7 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → dom 𝑥 = dom {⟨1𝑜, 1𝑜⟩})
42	40, 41	sseq12d 3831	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1𝑜, 1𝑜⟩} ⊆ dom {⟨1𝑜, 1𝑜⟩}))
43	39, 42	mpbiri 249	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 352	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 575	. . . 4 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 4846	. . . . . 6 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3174	. . . . 5 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 2012	. . . 4 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 270	. . 3 ⊢ (𝑥 = {⟨1𝑜, 1𝑜⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)))
50		ax-dc 9549	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3452	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1𝑜, 1𝑜⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 2022	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   = wceq 1637  ∃wex 1859   ∈ wcel 2156   ≠ wne 2978  ∀wral 3096  Vcvv 3391   ⊆ wss 3769  ∅c0 4116  {csn 4370  ⟨cop 4376   class class class wbr 4844  dom cdm 5311  ran crn 5312  suc csuc 5938  Fun wfun 6091  ‘cfv 6097  ωcom 7291  1_𝑜c1o 7785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7175  ax-dc 9549

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-ord 5939  df-on 5940  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-fv 6105  df-1o 7792

This theorem is referenced by:  axdc2lem  9551  axdc3lem  9553  axdc4lem  9558  axcclem  9560